mmtheorems79 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7801-7900 - Page 79 of 108

Type	Label	Description
Statement
Syntax	cbl 7801	Extend class notation with the metric space ball function.
class ball
Syntax	copn 7802	Extend class notation with a function mapping each metric space to the family of its open sets.
class Open
Definition	df-met 7803	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 7804. However, we will often also call the metric itself a "metric space.") Equivalent to Definition 14-1.1 of [Gleason] p. 223. See ismsg 7810 for the property "is a metric space." The 4 properties in Gleason's definition are shown by met0 7825, metgt0 7830, metsym 7826, and mettri 7827.
|- Met = {x | E.y(x:(y X. y)-->RR /\ A.z e. y A.w e. y (((zxw) = 0 <-> z = w) /\ A.v e. y (zxw) <_ ((vxz) + (vxw))))}
Definition	df-ms 7804	Define the (proper) class of all metric spaces.
|- MetSp = {<.x, y>. | (y e. Met /\ x = dom dom y)}
Definition	df-bl 7805	Define the metric space ball function. See blval 7847 for its value.
|- ball = {<.x, y>. | (x e. Met /\ y = {<.<.z, w>., v>. | ((z e. dom dom x /\ w e. RR) /\ (0 < w /\ v = {u e. dom dom x | (zxu) < w}))})}
Definition	df-opn 7806	Define a function whose value is the family of open sets of a metric space. See isopn 7869 for its main property.
|- Open = {<.x, y>. | (x e. Met /\ y = {z | (z (_ dom dom x /\ A.w e. z E.v e. ran ( ball ` x)(w e. v /\ v (_ z))})}
Theorem	msrel 7807	The class of all metric spaces is a relation.
|- Rel MetSp
Theorem	ismet 7808	Express the predicate "D is a metric."
|- X = dom dom D   =>   |- (D e. A -> (D e. Met <-> (D:(X X. X)-->RR /\ A.x e. X A.y e. X (((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy))))))
Theorem	dfms2 7809	Alternate definition for the class of all metric spaces (replaces old version of df-ms 7804).
|- MetSp = {<.x, y>. | (y:(x X. x)-->RR /\ A.z e. x A.w e. x (((zyw) = 0 <-> z = w) /\ A.v e. x (zyw) <_ ((vyz) + (vyw))))}
Theorem	ismsg 7810	Express the predicate "<.X, D>. is a metric space" with underlying set X and distance function D.
|- (D e. A -> (<.X, D>. e. MetSp <-> (D:(X X. X)-->RR /\ A.x e. X A.y e. X (((xDy) = 0 <-> x = y) /\ A.z e. X (xDy) <_ ((zDx) + (zDy))))))
Theorem	ismsi 7811	Properties that determine a metric space.
|- D e. V   &   |- D:(X X. X)-->RR   &   |- ((x e. X /\ y e. X) -> ((xDy) = 0 <-> x = y))   &   |- ((x e. X /\ y e. X /\ z e. X) -> (xDy) <_ ((zDx) + (zDy)))   &   |- M = <.X, D>.   =>   |- M e. MetSp
Theorem	ismeti 7812	Properties that determine a metric.
|- X e. V   &   |- D:(X X. X)-->RR   &   |- ((x e. X /\ y e. X) -> ((xDy) = 0 <-> x = y))   &   |- ((x e. X /\ y e. X /\ z e. X) -> (xDy) <_ ((zDx) + (zDy)))   =>   |- D e. Met
Theorem	msflem 7813	Lemma for msf 7814 and others.
Theorem	msf 7814	Mapping of the distance function of a metric space.
|- X = (1st` M)   &   |- D = (2nd` M)   =>   |- (M e. MetSp -> D:(X X. X)-->RR)
Theorem	mscl 7815	Closure of the distance function of a metric space.
|- X = (1st` M)   &   |- D = (2nd` M)   =>   |- ((M e. MetSp /\ A e. X /\ B e. X) -> (ADB) e. RR)
Theorem	metflem 7816	Lemma for metf 7817 and others.
Theorem	metf 7817	Mapping of the distance function of a metric space.
|- X = dom dom D   =>   |- (D e. Met -> D:(X X. X)-->RR)
Theorem	metdmdm 7818	The base set of a metric space in terms of its distance function.
|- X = dom dom D   =>   |- (D e. Met -> X = dom dom D)
Theorem	metssba 7819	The base set of a metric subspace.
|- X = dom dom D   =>   |- (D e. Met -> (X i^i Y) = dom dom ( D |` (Y X. Y)))
Theorem	metssba2 7820	The base set of a metric subspace.
|- X = dom dom D   =>   |- ((D e. Met /\ Y (_ X) -> Y = dom dom ( D |` (Y X. Y)))
Theorem	metcl 7821	Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> (ADB) e. RR)
Theorem	meteq0 7822	The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> ((ADB) = 0 <-> A = B))
Theorem	mettri2 7823	Triangle inequality for the distance function of a metric space.
|- X = dom dom D   =>   |- ((D e. Met /\ (C e. X /\ A e. X /\ B e. X)) -> (ADB) <_ ((CDA) + (CDB)))
Theorem	mettri4 7824	Triangle inequality for the distance function of a metric space.
|- X = dom dom D   =>   |- (((D e. Met /\ A e. X) /\ (B e. X /\ C e. X)) -> (ADB) <_ ((CDA) + (CDB)))
Theorem	met0 7825	The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X) -> (ADA) = 0)
Theorem	metsym 7826	The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> (ADB) = (BDA))
Theorem	mettri 7827	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223.
|- X = dom dom D   =>   |- ((D e. Met /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) <_ ((ADC) + (CDB)))
Theorem	mettri3 7828	Triangle inequality for the distance function of a metric space.
|- X = dom dom D   =>   |- ((D e. Met /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) <_ ((ADC) + (BDC)))
Theorem	metge0 7829	The distance function of a metric space is nonnegative.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> 0 <_ (ADB))
Theorem	metgt0 7830	The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse.
|- X = dom dom D   =>   |- ((D e. Met /\ A e. X /\ B e. X) -> (A =/= B <-> 0 < (ADB)))
Theorem	metne0 7831	A metric space is nonempty iff its base set is nonempty.
|- X = dom dom D   =>   |- (D e. Met -> (D =/= (/) <-> X =/= (/)))
Theorem	metreslem 7832	Lemma for metres 7833. (Contributed by FL, 10-Nov-2006.)
Theorem	metres 7833	A restriction of a metric is a metric.
|- (D e. Met -> (D |` (R X. R)) e. Met)
Theorem	metss 7834	If two metrics are in a subset relationship, so are their base sets.
|- X = dom dom C   &   |- Y = dom dom D   =>   |- (C (_ D -> X (_ Y)
Theorem	0met 7835	The empty metric.
|- (/) e. Met
Theorem	metxplem1 7836	Lemma for metxp 7844.
Theorem	metxplem2 7837	Lemma for metxp 7844.
Theorem	metxplem3 7838	Lemma for metxp 7844.
Theorem	metxpdval 7839	Value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   &   |- F = (1st` R)   &   |- G = (2nd` R)   &   |- H = (1st` S)   &   |- J = (2nd` S)   =>   |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = if((GCJ) < (FBH), (FBH), (GCJ)))
Theorem	metxptval 7840	One case of the value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}   &   |- F = (1st` R)   &   |- G = (2nd` R)   &   |- H = (1st` S)   &   |- J = (2nd` S)   =>   |- (((R e. (X X. Y) /\ S e. (X X. Y)) /\ (GCJ) <_ (FBH)) -> (RDS) = (FBH))
Theorem	metxpfval 7841	One case of the value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.
|- X = dom dom B   &   |- Y = dom dom C   &   |- B e. Met   &   |- C e. Met   &   |- D = {<.<.x, y